Question

What if L'hospital's rule doesn't work?

Answer 1

l'Hopital's Rule occationally fails by falling into a never ending cycle. Let us look at the following limit.

`lim_{n to infty}{sqrt{x^2+1}}/x`

by l'Hopital's Rule (`infty`/`infty`),

`=lim_{n to infty}{x/{sqrt{x^2+1}}}/{1}=lim_{n to infty}x/sqrt{x^2+1}`

by l'Hopital's Rule (`infty`/`infty`),

`=lim_{n to infty}1/{x/sqrt{x^2+1}}=lim_{n to infty}{sqrt{x^2+1}}/x`

As you can see, the limit came back to the original limit after applying l'Hopital's Rule twice, which means that it will never yield a conclusion. So, we just need to try another approach.

`lim_{n to infty}{sqrt{x^2+1}}/x`

by including the denominator under the square-root,

`=lim_{n to infty}sqrt{{x^2+1}/x^2}`

by simplifying the expression inside the square-root,

`lim_{n to infty}sqrt{1+1/x^2}=sqrt{1+1/infty^2}=sqrt{1+0}=1`

So, we could come up with the limit without using l'Hopital's Rule.

I hope that this was helpful.